Particle Propagation in the Galactic Center and Spatial Distribution of Non-Thermal X-rays
Vladimir Dogiel, Dmitrii Chernyshov, Takayuki Yuasa, Kwong-Sang Cheng, Aya Bamba, Hajime Inoue, Chung-Ming Ko, Motohide Kokubun, Yoshitomo Maeda, Kazuhisa Mitsuda, Kazuhiro Nakazawa, Noriko Y. Yamasaki
aa r X i v : . [ a s t r o - ph . H E ] J un Particle Propagation in the Galactic Center and SpatialDistribution of Non-Thermal X-rays
Vladimir
Dogiel , , Dmitrii Chernyshov , , Takayuki Yuasa , Kwong-Sang Cheng , Aya Bamba , Hajime Inoue , Chung-Ming Ko , Motohide Kokubun , Yoshitomo Maeda ,Kazuhisa Mitsuda , Kazuhiro Nakazawa , and Noriko Y. Yamasaki Institute of Space and Astronautical Science, 3-1-1, Yoshinodai, Sagamihara, Kanagawa, 229-8510,Japan P.N.Lebedev Institute, Leninskii pr, 53, 119991 Moscow, Russia, [email protected] Moscow Institute of Physics and Technology, Institutskii lane, 141700 Moscow Region,Dolgoprudnii, Russia Department of Physics, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,Tokyo 113-0033 Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong, China Institute of Astronomy, National Central University, Jhongli 320, Taiwan (Received 2000 December 31; accepted 2001 January 1)
Abstract
We showed that if the non-thermal emission from the Galactic center in the range14 −
40 keV is due to inverse bremsstrahlung emission of subrelativistic protons,their interactions with hot and cold fractions of the interstellar medium are equallyimportant. Our estimation show that about 30% of the total non-thermal flux fromthe GC in the range 14 −
40 keV is generated in regions of cold gas while the restis produced by proton interaction with hot plasma. From the spatial distribution of6.7 keV iron line we concluded the spatial distribution of hot plasma is strongly non-uniform that should be taken into account in analysis of protons propagation in theGC. From the Suzaku data we got independent estimates for the diffusion coefficientof subrelativistic protons in the GC, which was in the range 10 − cm s − Key words:
Galaxy: center - X-rays: diffuse background - ISM: cosmic rays
1. Introduction
This is a final paper of the series (Cheng et al. 2006; Cheng et al. 2007; Dogiel et al.2008; Dogiel et al. 2009abc) on energetic processes in the Galactic center (GC). In these paperswe presented our interpretation of X-ray and gamma-ray emission from the Galactic center.1e supposed that all these phenomena had common origin and were initiated by accretionprocesses onto the central supermassive black hole.We showed that the the observed X-ray continuum and line emission from the GC mightbe produced by a flux of subrelativistic protons which resulted from an unbounded part of starsaccreted onto the central black hole. We estimated the average energy of escaping protons tobe about 100 MeV in order to produce a flux of hard X-ray emission in the energy range14 −
40 keV as observed by Suzaku/HXD (Yuasa et al. 2008).The quasi-stationary production rate of subrelativistic protons hardly exceeds Q ∼ × protons s − for the frequency of star capture ∼ − − − years − (Syer & Ulmer 1999;Donley et al. 2002) and a fraction of escaped matter equaled ∼
50% of star masses (Ayal et al.2000). The unbounded fraction of stars escapes with subrelativistic velocities which correspondenergies about 100 MeV for protons and 50 keV for electrons (see Dogiel et al. 2009a). Thenumbers of protons and electrons equal to each other. Therefore, fluxes of bremsstrahlung hardX-ray emission produced by protons and by electrons equal to each other too. But the lifetimeof electrons with energies 50 keV is in about five orders of magnitude smaller than that of100 MeV protons and the electron bremsstrahlung flux is significant for a short time just aftera capture event. Since it is supposed that relatively long time has passed after the last starcapture, electron bremsstrahlung radiation is negligible.Our goal is to estimate the spatial diffusion coefficient, D , of subrelativistic particlesnear the GC, whose value is unknown. Below we derive its value from the spatial distributionof hard X-ray emission near the GC as observed by Suzaku.In section 2, we summarize an energetics of the Galactic Center diffuse emission. Insection 3, we carry forward our model using non-uniform target gas distribution. Section 4 isdevoted to explain how the model and the HXD data can be compared. In section 5 and 6,spatial distribution of 6.7 keV Fe line emission and hard X-ray continuum are compared withones expected from the present model, deriving a confinement on a diffusion coefficient for thesub-relativistic protons.
2. Components of Hard X-Ray Emission from the GC in the range 14 −
40 keV
Koyama et al. (2007b) reported that the hard X-ray spectrum in the range 2 −
10 keVfrom the GC is naturally explained by a 6.5 keV-temperature plasma plus a power-law compo-nent with the photon index of Γ = 1 .
4. Latter on, Yuasa et al. (2008) found a prominent hardX-ray emission in the range from 14 to 40 keV whose spectrum is a power law with the spectralindex ranging from 1.8 to 2.5.Different processes may contribute to the total emission from the GC.
Thermal emission . The emissivity of thermal bremsstrahlung of a hot plasma ε can beestimated from the equation (see e.g. Ginzburg 1989)2 πε = 1 . × − n ( r ) q T ( r ) erg cm − s − (1)where the temperature is in Kelvin degrees.As follows from the analysis of Suzaku data the plasma temperature in the GC isconstant and equals 6.5 keV (Koyama et al. 2007b). Then the thermal emissivity in the GC is4 πε ≃ . × − n ( r ) erg cm − s − (2)Dogiel et al. (2009b) presented results of calculation for the 0 . ◦ × . ◦ central regionand showed that for the plasma density n = 0 . − and the temperature 6.5 keV the totalthermal flux from this region in the energy rage 14 −
40 keV was about F th ≃ × erg s − .The total non-thermal flux was estimated by the value F nth ≃ × erg s − . Non-thermal component . Koyama et al. (2009) found that a combination of the Fe XXV-K α and Fe I-K α fluxes was proportional to the non-thermal continuum flux in the 5 - 10 keVenergy band. They concluded that the total non-thermal flux consists of the two components,one of which is proportional to the intensity of 6.7 keV line, and, therefore, is produced inregions of hot plasma, while the intensity of the second component is proportional to that ofthe 6.4 keV line and is generated in regions of molecular gas. Warwick et al. (2006) reported asimilar conclusion obtained from the XMM-Newton observations.In the model of X-ray production by subrelativistic protons this relation is naturallyexplained (see Dogiel et al. 2009bc). Continuum and line emission is produced in this case byinteractions of subrelativistic protons with the background plasma and the molecular hydrogenin the GC.In spite of relatively small radius ( r ∼
200 pc) the inner Galactic region contains about10% of the Galaxy’s molecular mass, ∼ M H ≃ (7 − × M ⊙ . Most of the molecular gas iscontained in very compact clouds (see the review of Mezger et al. 1996). Since the total massof the molecular gas is much larger than that of the hot plasma, M pl ∼ × M ⊙ , one mayassume that most of inverse bremsstrahlung flux produced by protons would come from regionsfilled by the molecular gas. However, as Dogiel et al. (2009c) showed, the mean free path ofprotons in dense molecular clouds is very short since the diffusion coefficient inside molecularclouds is quite small. Therefore, only a small fraction of the molecular hydrogen is involvedinto processes of inverse bremsstrahlung radiation. Thus, for the molecular cloud Sgr B2 withthe mass about 10 M ⊙ it follows from calculations of Dogiel et al. (2009c) and the Suzakudata (see Koyama et al. 2007a) that the total bremsstrahlung flux in the range 2 −
10 keV isabout F − ≃ erg s − . The total flux of the 6.4 keV line emission from Sgr B2 is about F . ∼ . × erg s − that gives the ratio F − /F . ∼
8. For higher energies of X-ray photonsour calculations show that the ratio of F − /F . ∼ . . ◦ × . ◦ it is about ∼ . × erg s − , then we expectthat the total non-thermal bremsstrahlung flux from proton interaction with the molecular gas3s about 10 erg s − , and 70% of the total non-thermal bremsstrahlung emission in the range14 −
40 keV, ∼ × erg s − , is generated in regions filled with the hot plasma.
3. Qualitative effect of the plasma density variations
In this section we present a qualitative effect of plasma density variations on the protondistribution in the GC. We remind that the spatial distribution of subrelativistic protons isdescribed by the quasi-stationary diffusion equation ∂∂E ( b ( E ) N ) − ∇ D ∇ N = Q ( E, r ) , (3)where N is the density of cosmic rays, D is the spatial diffusion coefficient, dE/dt ≡ b ( E ) is therate of energy losses which for subrelativistic protons is determined by the Coulomb collisionsand has the form dEdt ! i = − πne ln Λ m v p ≃ k √ E , (4)where v p is the proton velocity, ln Λ is the Coulomb logarithm, and n is the plasma density,which is a function of the radius r in general case. The source function is taken in the form Q ( E ) = Qδ ( E − E esc ) δ ( r ) . (5)where Q ≃ × protons s − and E esc ≃
100 MeV is the injection energy of protons.For n = n = const , the spherically symmetric solution is N ( r, E ) = Q | b ( E ) | (4 πλ ) / exp " − r λ (6)where λ ( E ) = E Z E D ( E ) b ( E ) dE = 2 D k (cid:16) E / − E / (cid:17) (7)In the case when the plasma density is spatially variable as n ( r ) = n (cid:18) r r (cid:19) , (8)where r is a scale parameter, the analytical solution for N ( r, E ) is N ( r, E ) = Q | b ( E ) | r (4 πλ ) / r r r exp " − λ ( E ) r exp " − ln ( r/r ) r λ ( E ) (9)Here λ = 14 E Z E D ( E ) b ( E ) dE (10)In figure 1 we show the difference between uniform for n = 0 . − (solid line) and non-uniform cases for the parameters n = 10 cm − (dashed-dotted line) and n = 3 cm − (dashedline). The scale parameter is taken to be r = 25 pc. One can see that variations of the gasdensity change significantly the proton distribution.4 H pc L ´ - ´ - N H cm - L Fig. 1.
The spatial distribution of 70 MeV protons in the GC for the case of uniform distribution (solidline) and that of non-uniform distribution equation (9) for n = 10 cm − (dashed-dotted line) and n = 3cm − (dashed line).
4. How to compare the model calculation and the observed HXD data?
In section 5 and 6, we will compare spectra expected from the present model with theHXD spectrum observed around the GC (Yuasa et al. 2008). Before describing the comparisonresult, we present methods and assumptions which were used in the calculation.The intensity I of inverse bremsstrahlung emission of protons in any direction s is cal-culated from I ( E x , s ) = 14 π Z s n ( r ) ds Z E N ( E, r ) v dσ ib dE x dE (11)where the integration is along the line of sight s and E x is the energy of photons. Here N ( E, r )is the density of subrelativistic protons with the energy E at the radial distance r from theGC, v is the velocity of protons with the energy E , dσ ib /dE x is the cross-section of inversebremsstrahlung radiation, and n ( r ) is the plasma density distribution in the GC.To compare such results of our model calculations, which are given in photonscm − keV − s − sr − with the Suzaku data of Yuasa et al. (2008) given in counts s − , we shouldconvolve the HXD energy and angular responses (see Mitsuda et al. 2007; Takahashi et al.2007; Kokubun et al. 2007) to the model intensity. The procedure is as follows: f − = Z ℓ,b d Ω Z dEA ( ℓ, b ) S ( E ) I ( E, ℓ, b ) , (12)where f is in ( counts s − ) denotes the expected count rate in the direction determined bythe galactic coordinates ( ℓ, b ), Ω is in ( sr ), I ( E, ℓ, b ) in ( ph cm − keV − s − sr − ) is the contin-uum intensity in the direction ( ℓ, b ), S ( E ) and A ( ℓ, b ) represent the effective area and angulartransmission of the HXD, respectively. 5he HXD data can still include a contaminating flux from a number of X-ray pointsources besides the diffuse emission as noted in Dogiel et al. (2009b). The flux level of thecontamination is calculated to be on the order of 10% by integrating the known LogN − LogScurve for X-ray point sources around the GC obtained with Chandra (Muno et al. 2009) overthe luminosity of of 2 × − × erg s − in which range has been actually measured sofar (in the ”field” region of Muno et al. 2009). Although the contribution can be larger if theLogN − LogS curve is measured further smaller luminosities, in the present comparison with themodel calculation, we simply ignore their contribution. The effect of this neglection is discussedlater in section 6.
5. Spatial Variations of the Gas Density in the GC
The spatial distribution of the hot plasma in the GC can be derived from the distributionof the 6.7 keV iron line which traces the hot plasma.The origin of the K- α iron line from the hot plasma of the GC is pure thermal (seeDogiel et al. 2009b). Then for the surface brightness distribution of the 6.7 keV emission linein any direction s to the GC we have I . ( s ) ∝ Z s n ( r ) ds (13)Maeda (1998) obtained a surface brightness distribution of the 6.7 keV emission line, I . , from the ASCA data as a function of the angle from the Galactic center, θ , I . ( ℓ, b ) = I exp( − | θ | ω ) + I exp( − | θ | ω ) (14)cos θ = cos ℓ cos b where θ is the angle from ( l, b ) = (0 , I =19.7 and I =1.6 in photons cm − s − sr − units, and ω = 0 .
42 and ω = 15 in degrees.This distribution is shown in figure 2 by the thick solid line.From the 6.7 line distribution of Maeda (1998) we try to derive the plasma densitydistribution in the GC. Then we will use this density distribution for calculations of hard X-rayemission from the GC. We represent the hot plasma distribution by analytical functions as n ( r ) = n exp (cid:20) − (cid:18) rr (cid:19) α (cid:21) , (15)and our goal is to find suitable parameters r and α which give good correspondence of theobserved ASCA data and that obtained with the density distribution (15).In order to derive the model spatial variations of the 6.7 keV line emission along theGalactic plane we should integrate along the direction s shown in figure 3. The angle betweenthe direction to the GC and the line of view is the Galactic longitude ℓ . The circle line aroundthe Galactic center defines a sphere filled with a hot plasma whose radius is supposed to beabout a = 200 pc in accordance with the X-ray observations (see, e.g. Koyama et al. 1996).6 .2 0.4 0.6 0.8 1 Θ H degree L Lg @ Y H phcm - s - sr - LD Fig. 2.
The latitude distribution of 6.7 keV line as from the ASCA data byMaeda (1998) is shown by thesolid line. Model spatial distribution of the surface brightness profile of the 6.7 keV emission line expectedfrom equation (17) for plasma distributions: r = 25 pc and α = 0 . Fig. 3.
The schematic view of the GC from Earth. The arrow line shows the distance between Earth andthe GC, the solid line is the line of view, the solid circle shows the region of hot plasma around the GC;the two radius-vectors mark positions of ψ max and ψ min = π − ψ max . The radius-vector r from the GC along the line of sight as a function of the Galactic longitude ℓ and the angle ψ between the line coinciding Earth an the GC (arrow line) is described as r ( ℓ, ψ ) = d · sin ℓ cos ψ (16)Then with the density distribution (15) the integration along the line of sight in the direction s is I . ∝ ψ max Z π − ψ max n ( r ( ℓ, ψ )) dsdψ dψ (17)where dsdψ = − a · cos ψ max cos ψ (18)7nd the angles ± ψ max shown by the two radius-vectors in figure 3 is ψ max = arccos " da sin ℓ (19)Here d = 8 kpc is the distance between Earth and the GC which equals 8 kpc (Reid 1993). Fromequation (17) and the ASCA 6.7 keV spatial variations obtained by Maeda (1998) we derivedparameters r and α , that gives r = 25 −
75 pc and α = 0 . − .
7. The ASCA spatial variationsof 6.7 keV line are shown in figure 2 by the thick solid line. As an example we show also inthis figure the model spatial variations of the 6.7 keV line intensity derived for r = 25 pc and α = 0 . n of equation (15) can be roughly estimated by comparing the modelemission (6.5 keV thermal + non-thermal) with the observed spectrum of the hard X-rayemission around the GC (Yuasa et al. 2008). To perform such a comparison, we convolvedthe model spectrum with the HXD response as described in section 4, and then, overlaid itover the HXD spectrum by adjusting the model normalization factor, n in equation (15), sothat the model reproduces the HXD fluxes. Figure 4(a) shows a result of the convolution, andthe model well reproduce the observed spectrum in the 14 −
40 keV band. Although we onlypresent the result for one parameter set explained in the caption of figure 4, the model spectralshape does not strongly depend on the parameters, and within possible parameter ranges, n was calculated to be ∼ . − . − .As one can see from these figures the contributions of thermal and non-thermal emissioninto the total flux from the GC are almost equal to each other in this energy range (their ratiois 2:3).
6. Diffusion Coefficient of Cosmic Rays in the Galactic Center
In order to calculate the spatial distribution of subrelativistic protons in the GC weshould take into account energy losses in regions of hot plasma and in the regions of neutralgas surrounding the GC whose average density was taken to be n H = 1 cm − .As follows from the analysis of Yuasa et al. (2008) the spatial distribution of the intensi-ties of thermal and non-thermal emission in the GC are similar to each other. Since the first oneis proportional I th ∝ R r n ( r ) ds and the second one is proportional to I nth ∝ R r n ( r ) N p ( r ) ds , where N p ( r ) is the spatial distribution of subrelativistic protons, it means that the spatial distributionof subrelativistic protons does not differ strongly from that of plasma.The expected spatial distributions of the 14 −
40 keV X-ray emission in the GC derivedfrom equation (11) for D = 3 × cm s − (solid line) and for D = 3 × cm s − (dashed line)8 nergy (keV)10 20 30 40 50 ) - k e V - C oun t R a t e ( C oun t s s -5 -4 -3 -2 -1
10 Energy (keV)10 20 30 40 50 ) - ( P I N F OV ) - k e V - s - P ho t on F l ux ( P ho t on s c m -5 -4 -3 Fig. 4. (a)The 14 −
40 keV X-ray spectrum observed by HXD/PIN around the GC (taken from Yuasaet al. 2008). The model spectra for 6.5 keV thermal emission, inverse bremsstrahlung, and sum of themare also shown in dashed, dashed-dotted, and solid line. The model spectra were calculated using suchparameters as r = 50 pc, α = 1 . D = 1 × cm /s, and n = 0 .
13 cm − . (b)The same model spectra as(a) are shown without convolving the HXD/PIN energy response but with integration over the PIN FOV(0 . ◦ × . ◦
55, effectively). and the Suzaku data obtained by Yuasa et al. (2008) are shown in figure 5 . Our calculationsshow that spatial distribution of the non-thermal 14 −
40 keV X-ray emission does not differsignificantly from each other. However, the necessary proton production rate Q is an essentialfunction of the diffusion coefficient. Thus, for D = 3 × cm s − the value of Q needed toreproduce the Suzaku data equals 10 protons s − while for D = 3 × cm s − the productionrate is necessary to be Q = 2 . × protons s − . We remind that the average rate of protonproduction by accretion on the black hole is no more than Q = 3 × protons s − . Since thevalue of Q is restricted, then permitted values of D are confined within a relatively narrowrange around D ≃ cm s − .However, these estimates were obtained in assumption that almost 100% of the 14 −
40 keV X-ray flux from the GC is generated by subrelativistic protons. As follows from, e.g.,Warwick et al. (2006); Revnivtsev et al. (2009) a significant part of the GC hard X-ray fluxmay be due to faint point sources. Then a smaller production rate of protons in the GC isrequired, and, hence, higher values of the diffusion coefficient cannot be excluded. In figure 5, the data point at l = 0 . ◦
55 is largely discrepant from the model, and even from the other datapoints. As noted in Yuasa et al. (2008), this deviation could be due to underestimation of contaminatingsignals from known bright point source inside the HXD/PIN FOV. Therefore, this point should be ignoredin the present comparison between observed data and the model calculation. −1 longitude (degree) C oun t s s − Fig. 5.
The spatial distribution of inverse bremsstrahlung flux in the energy range 12 −
40 keV averagedover the HXD field of view with the Suzaku data. Solid line: D = 3 × cm s − , Q = 10 protons s − .Dashed line: D = 3 × cm s − , Q = 2 . × protons s − . The Suzaku data were taken from Yuasaet al. (2008)
7. Discussion and Conclusion
Analysis of X-ray data (Koyama et al. 2009) demonstrated that continuum non-thermalemission in the GC consisted of two components one of which was proportional to the intensityof 6.7 keV iron line which traced distribution of hot 6.5 keV plasma, and the other one wasproportional to the intensity of 6.4 keV plasma which traced distribution of cold molecularhydrogen. If this non-thermal emission is due to inverse bremsstrahlung emission of subrela-tivistic protons, then their interactions with hot and cold fractions of the interstellar mediumare equally important. Our estimation show that about 30% of the total non-thermal flux fromthe GC in the range 14 −
40 keV is generated in regions of cold gas while the rest is producedby proton interaction with hot plasma. The reason is that only a small fraction of moleculargas is involved into processes of bremsstrahlung emission because of short mean free path ofprotons in molecular clouds.From the X-ray data obtained with ASCA (Maeda 1998) it was concluded that theplasma density distribution in the GC is strongly non-uniform, e.g. the model distribution of6.7 keV line in the GC obtained with the plasma profile n ( r ) = 0 . × exp − r ! . (cm − ) (20)is very close to that from ASCA observation. This effect of density variations should be includedinto analysis of protons propagation in the GC.From the total spectrum of X-ray emission in the range 14 −
40 keV from the 0 . ◦ -10iameter central region and the spatial variation of this emission within 1 . ◦ -diameter centralregion as observed by Suzaku and ASCA we derived the characteristic diffusion coefficient ofsubrelativistic protons. It was shown that the diffusion coefficient of subrelativistic particlesis about the value of 3 · cm s − . For higher values of the diffusion coefficient a higherproduction rate of subrelativistic protons is necessary which cannot be provided by accretion.However, this estimation of D one should consider as a lower limit. The point is that we assumedhere that all hard X-ray emission in the 14 −
40 keV band is produced by subrelativistic protonsin the hot fraction of the interstellar gas. However as stated above a part of this emission maybe produced in molecular clouds. Besides, we do not know exactly which part of this emissionis due to unresolved point sources (see discussion in Dogiel et al. 2009b). Therefore, we cannotexclude that the diffusion coefficient in the GC is close to its average value in the Galaxy, i.e. D ∼ cm s − . Thus, we determine the range of values of the diffusion coefficient in the GCas 10 − cm s − .The authors are grateful to the referee, Prof. Fumiaki Nagase, for his comments andcorrections.VAD and DOC were partly supported by the RFBR grant 08-02-00170-a, the NSC-RFBR Joint Research Project No 95WFA0700088 and by the grant of a President of theRussian Federation ”Scientific School of Academician V.L.Ginzburg”. KSC is supported by aRGC grant of Hong Kong Government under HKU 7014/07P. A. Bamba is supported by JSPSResearch Fellowship for Young Scientists (19-1804). CMK is supported in part by NationalScience Council, Taiwan under the grant NSC-96-2112-M-008-014-MY3. References
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